An object of mass M and radius R rolls without slipping down a fixed incline that makes an angle θ with the horizontal. Its rotational inertia about the center of mass can be written in the form I=kMR2, where k is a dimensionless constant that depends on the mass distribution.
(a) Derive an expression for the magnitude of the linear acceleration a of the center of mass in terms of g, θ, and k.
(b) Three objects are released from rest at the same height on identical inclines: a solid sphere (k=52), a solid cylinder (k=21), and a thin hoop (k=1). Use the result from part (a) to rank their accelerations. Clearly indicate any ties and justify your ranking.
(c) For each of the three objects in part (b), determine an algebraic expression for the ratio KtotKtrans at any instant during the descent. Rank the objects according to the magnitude of this ratio and explain what the ranking implies about how each object partitions its gravitational potential energy.